Generative deep learning systems offer powerful tools for artefact generation, given their ability to model distributions of data and generate high-fidelity results. In the context of computational creativity, however, a major shortcoming is that they are unable to explicitly diverge from the training data in creative ways and are limited to fitting the target data distribution. To address these limitations, there have been a growing number of approaches for optimising, hacking and rewriting these models in order to actively diverge from the training data. We present a taxonomy and comprehensive survey of the state of the art of active divergence techniques, highlighting the potential for computational creativity researchers to advance these methods and use deep generative models in truly creative systems.